A Characterization of Curves that Lie on a Totally Umbilical Surface of a Space Form

Abstract

We give necessary and sufficient conditions on the curvature and the torsion of a regular curve of the space forms $\h^3$ and $\s^3$ to be contained in a totally umbilical surface. In case that the curve has constant torsion, we obtain the value of the curvature of the curve. Also numerical pictures of these curves are shown.

17 pages, 3 figures